Let be a lattice and let be a point in the upper half-plane. We prove the existence and give an explicit formula for the pair correlation density function for the set of angles between geodesic rays of the lattice intersected with increasingly large balls centered at , thus proving a conjecture by Boca, Popa, and Zaharescu.
Dubi Kelmer. Alex Kontorovich. "On the pair correlation density for hyperbolic angles." Duke Math. J. 164 (3) 473 - 509, 15 February 2015. https://doi.org/10.1215/00127094-2861495